I Superposition of Phases: A Key Concept in Connecting QFT and Statistical Physics

[Heidi]

Homework Statement: relation between qft and statistical physics
Homework Equations: domains with equal values

I read a french paper about Kenneth Wilson.
i translate several sentences (with google):

it was demonstrated in 1960 by Kenneth Wilson that renormalization formed an incongruous bridge between statistical physics and quantum field theory.
He connected the behavior of a system during a transition second-order phase to the behavior of a QFT field. More specifically, during the second-order phase transition, two phases of matter overlap, which leads to
scale invariance at the critical point.

My question is about this overlapping. the french word is "superpose"

How can phases overlap or be superposed? superposition is used in qft but here what does it mean?

 

[Mentz114]

Homework Statement: relation between qft and statistical physics
Homework Equations: domains with equal values

I read a french paper about Kenneth Wilson.
i translate several sentences (with google):

it was demonstrated in 1960 by Kenneth Wilson that renormalization formed an incongruous bridge between statistical physics and quantum field theory.
He connected the behavior of a system during a transition second-order phase to the behavior of a QFT field. More specifically, during the second-order phase transition, two phases of matter overlap, which leads to
scale invariance at the critical point.

My question is about this overlapping. the french word is "superpose"

How can phases overlap or be superposed? superposition is used in qft but here what does it mean?

The word 'phase' in statistical physics can mean a phase of matter such as solid, liquid, vapour or plasma. Naturally in QFT it means something different.

Different phases of matter can co-exist at their boundary and this is sometimes called a superposition. See for instance Rao, Statistical Physics and Thermodynamics, Oxford ( 2017), chapter 8.

 

[Heidi]

thanks. I can understand that there are domains separated by walls in which the phases overlap. What is happening a the critical point for the walls and the domains?

 

[Demystifier]

What is happening a the critical point for the walls and the domains?

It is greatly illustrated in this video:


(I have found the link to this video in the lectures by Tong on Statistical Field Theory.)

 

[Heidi]

Is it a simulation on a computer?

 

[king vitamin]

More specifically, during the second-order phase transition, two phases of matter overlap, which leads to
scale invariance at the critical point.

I think even at best this is very imprecise phrasing. It is often useful to think of a first-order or discontinuous phase transition as a region where two (generically very different) phases coexist. These transitions are not scale invariant and not described by QFT or RG, but I would say that in these systems two phases of matter are overlapping.

In contrast, as continuous or critical phase transitions, the thermodynamic properties of the system change continuously (but still with non-analyticities), so this only happens between phases that are somehow "similar." I like to think of the critical state to be like a phase in itself (and in fact there are stable phases with scale invariance). It's kind of like a balancing act between two very similar phases - see the nice video posted by Demystifier.

The word 'phase' in statistical physics can mean a phase of matter such as solid, liquid, vapour or plasma. Naturally in QFT it means something different.

I've never encountered a difference between the use of the word in those two fields. After all, QFT is one of the primary tools used in statistical physics to study criticality, and it is also common to consider QFTs at finite temperature where one needs statistical physics anyways.

 

[Mentz114]

[..]

Naturally in QFT it means something different.

I've never encountered a difference between the use of the word in those two fields. After all, QFT is one of the primary tools used in statistical physics to study criticality, and it is also common to consider QFTs at finite temperature where one needs statistical physics anyways.

In the text "Quantum Field Theory" by Itzykson & Zuber (1985) the word 'phase' has 3 entries in the index referring to 'phase space' and 'phase shifts'.

 

[king vitamin]

In the text "Quantum Field Theory" by Itzykson & Zuber (1985) the word 'phase' has 3 entries in the index referring to 'phase space' and 'phase shifts'.

Yes, the word is a homonym (and any statistical physics book will discuss phase space too). I'm not familiar with that textbook; it's a shame that it doesn't treat such an interesting aspect of quantum field theory. Perhaps I can recommend the following excellent textbooks which all treat phase diagrams and phase transitions of quantum field theories in detail:

Peskin & Schroeder - An Introduction to Quantum Field Theory
Zinn-Justin - Quantum Field Theory and Critical Phenomena
Polyakov - Gauge Fields and Strings
Kleinert - Critical properties of $\phi^4$ theories

 

[Heidi]

I wonder what is to be seen in the video given by Demystifier.
In the middle or the screen we have the case of the critical point. there are fluctuations at every scale. but fluctuations of what? At a given moment when water is boiling i can see gas bubbles with various sizes.
But at the critical point things become opalescent. gas ans liquid have the same density. So what are the domains in the video at the critical point (and at a greater temperature in the 3d screen)?

 

[Demystifier]

Is it a simulation on a computer?

Yes.

 

[Heidi]

can we say that the simulation shows the shape of the domains in the three screens with artistic contrast between them even when the difference of densities between gas and liquid tends to zero?

 

[Demystifier]

can we say that the simulation shows the shape of the domains in the three screens with artistic contrast between them even when the difference of densities between gas and liquid tends to zero?

I guess we can.

 

[Heidi]

it is possible that the sentence written by the author means that when we look at an equal volume in the two phases we see (at Tc) the same thing. same percentage for every lenth of domains etc. they can be superposed in this sense (they can be swapped).